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On nonlinear resonance oscillations of a spring supported point particle

Mikhail Shunderyuk
Full research of flat small nonlinear oscillations
of a spring pendulum with nonlinear dependence
of a tension of a spring on its lengthening is conducted.
The method of a hamiltonian normal form is used. For
reduction to a hamiltonian normal form the method of
invariant normalisation is used, what essentially reduces
calculations. Solutions of the normal form equations
have shown that periodic reorganisation between vertical
and horizontal oscillations occurs only in case of
resonances 1:1 and 2:1. At a resonance 2:1 this effect
is shown in square-law members of the equation, and
at a resonance 1:1 one should take into account cubic
members. In all other cases, both in the presence of
a resonance, and at its absence, oscillations have constant
frequencies with a little different from frequencies
of linear approach. For a resonance 2:1 it is found
maximum detuning of frequencies at which the effect
of swapping of energy from one kind of oscillation to
another disappears. The resonance 1:1 is physically
possible only for a spring possessing the negative cubic
term in the law of deformation.
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